Discrete Isoperimetry: Problems, Results,Applications and Methods

Gil KalaiHebrew University of Jerusalem

FPSAC’ 12, Nagoya, July 2012.

Gil Kalai Discrete Isoperimetry

The Isoperimetric Inequality

The Isoperimetric Inequality: The solid of volume V thatminimizes the surface area in Euclidean 3-space is the ball.

A sleeping, curled-up sphynx cat called Uthello. Author : SunnyRipert. Idea: Soul physics blog.

Gil Kalai Discrete Isoperimetry

The Isoperimetric Inequality

The Isoperimetric Inequality: The solid of volume V thatminimizes the surface area in Euclidean 3-space is the ball.

Gil Kalai Discrete Isoperimetry

Combinatorial Isoperimetric Relations - some examples

I (Erdos-Hanani:) A simple graph with m edges that minimizesthe number of vertices is obtained from a complete graph byadding a vertex and some edges.

I The Kruskal-Katona Theorem.

I Macaulay’s theorem.

I Harper’s theorem.

I A theorem of Descartes: A simple planar graph with m edgeshas at least m/3 + 2 vertices.

Gil Kalai Discrete Isoperimetry

Prelude: Baby Harper and canonical paths

Theorem: For a subset A of the discrete n-cube with |A| ≤ 2n−1

there are at least |A| edges from A to its complement.

Proof (hint): Consider a canonical path between two 0-1 vectorsx and y by flipping the disagreeing-coordinates from left to right.Then look at all canonical paths from all x ∈ A and y /∈ A. Thereare |A|(2n − |A|) such paths and each of them must contain anedge from A to its compliment.

On the other hand, every edge is contained in at most 2n−1

canonical paths.

Gil Kalai Discrete Isoperimetry

Part I: Isoperimetry, harmonic analysis, and probability

The first part of this lecture is about harmonic analysis applied todiscrete isoperimetry. We have several application and potentialapplications in mind mainly to problems in probability. I will startby mentioning one potential application. It deals with the theory ofrandom graphs initiated by Erdos and Renyi, and the model

G (n.p). In the picture we see arandom graph with n = 12 and p = 1/3.

Gil Kalai Discrete Isoperimetry

Threshold and Expectation threshold

Consider a random graph G in G (n, p) and the graph property: Gcontains a copy of a specific graph H. (Note: H depends on n; amotivating example: H is a Hamiltonian cycle.) Let q be theminimal value for which the expected number of copies of H ′ in Gis at least 1/2 for every subgraph H ′ of H. Let p be the value forwhich the probability that G contains a copy of H is 1/2.

Conjecture: [Kahn, K. 2006]

p/q = O(log n).

The conjecture can be vastly extended to general Booleanfunctions, and we will hint on possible connection with harmonicanalysis and discrete isoperimetry. (Sneak preview: it will require afar-reaching extension of results by Friedgut, Bourgain andHatami.)

Gil Kalai Discrete Isoperimetry

The discrete n-dimensional cube and Boolean functions

The discrete n-dimensional cube Ωn is the set of 0-1 vectors oflength n.

A Boolean function f is a map from Ωn to 0, 1.

A boolean function f is monotone if f cannot decrease when youswitch a coordinate from 0 to 1.

Gil Kalai Discrete Isoperimetry

The Bernoulli measure

Let p, 0 < p < 1, be a real number. The probability measure µp isthe product probability distribution whose marginals are given byµp(xk = 1) = p. Let f : Ωn → 0, 1 be a Boolean function.

µp(f ) =∑x∈Ωn

µp(x)f (x) = µpx : f (x) = 1.

Gil Kalai Discrete Isoperimetry

The total influence

Two vectors in Ωn are neighbors if they differ in one coordinate.

For x ∈ Ωn let h(x) be the number of neighbors y of x such thatf (y) 6= f (x).The total influence of f is defined by

I p(f ) =∑x∈Ωn

µp(x)h(x).

If p = 1/2 we will omit p as a subscript or superscript.

Gil Kalai Discrete Isoperimetry

Russo’s lemma

Russo’s lemma: For a monotone Boolean function f ,

dµp(f )/dp = I p(f ).

Very useful in percolation theory and other areas.

The threshold interval for a monotone Boolean function f isthose values of p so that µp(f ) is bounded away from 0 and 1.(Say 0.01 ≤ µp(f ) ≤ 0.99.)

A typical application of Russo’s lemma: If for every value p in thethreshold interval I p(f ) is large, then the threshold interval itself isshort. This is called a sharp threshold phenomenon.

Gil Kalai Discrete Isoperimetry

(A version of) Harper’s theorem

Harper’s theorem: If µp(f ) = t then

I p(f ) ≥ 2t · logp t.

There is a 3 line proof by induction.Harmonic analysis proof: without the log(1/t) factor it followsfrom Parseval.

Gil Kalai Discrete Isoperimetry

Influence of variables on Boolean functions

Let

σk(x1, . . . , xk−1, xk , xk+1, . . . , xn) = (x1, . . . , xk−1, 1−xk , xk+1, . . . , xn).

The influence of the kth variable on a Boolean function f isdefined by:

I pk (f ) = µp(x ∈ Ωn, f (x) 6= f (σk(x))).

Gil Kalai Discrete Isoperimetry

KKL’s theorem

Theorem (Kahn, K, Linial, 1988; Bourgain Katznelson KKL 1992;Talagrand 1994 Friedgut K. 1996) There exist a variable k suchthat

I pk (f ) ≥ Cµp(f )(1− µp(f )) log n/n.

A sharp version (due to Talagrand)∑I pk (f )/ log(e + I p

k (f )) ≥ C (p)µp(f )(1− µp(f )).

Gil Kalai Discrete Isoperimetry

Discrete Fourier analysis

We assume now p = 1/2. Let f : Ωn → R be a real function.

Let f =∑

f (S)WS be the Fourier-Walsh expansion of f .

HereWS(x1, x2, . . . , xn) = (−1)

Pxi :i∈S.

Gil Kalai Discrete Isoperimetry

Hypercontructivity and Harper’s theorem:

We assume now p = 1/2. f =∑

f (S)WS is the Fourier-Walshexpansion of f . Key ideas:

0 Parseval gives I (f ) = 4∑

f 2(S)|S |.1 Bonami-Gross-Beckner hypercontractive inequality.

||∑

f (S)(1/2)|S | || 2 ≤ || f || 5/4.

2 For Boolean functions the qth power of the q norm is themeasure of the support and does not depend on q. If thesupport is small this means that the q-norm is very differentfrom the r -norm if r 6= q.

(See also : Ledoux’ book on concentration of measure phenomena)

Gil Kalai Discrete Isoperimetry

Part II: Harper’s theorem: Stability and Symmetry

Gil Kalai Discrete Isoperimetry

Low Influence and Juntas

A dictatorship is a Boolean function depending on one variable. AK -junta is a Boolean function depending on K variables.

Theorem: (Friedgut, follows easily from KKL) If p is boundedaway from 0 and 1 and I p(f ) < C then f is close to a K (C )-Junta.

This works if log p/ log n = o(1) the most interesting applicationswould be when p is a power of n. There the theorem is not true.

Early stability results for Harper’s theorem were obtained in the70s by Peter Frankl.

Gil Kalai Discrete Isoperimetry

The works of Friedgut and Bourgain (1999)

Suppose that f is a Boolean function and

I p(f ) < pC ,

thenFriedgut’s theorem (1999): If f represent a monotone graphproperty then f is close to a a “locally defined” function g .

Bourgain’s theorem (1999): Unconditionally, f has a substantial“locally defined” ingredient.

Gil Kalai Discrete Isoperimetry

Hatami’s theorem: Pseudo-juntas

Suppose that for every subset of variables S , we have a functionJS : 0, 1S → 0, 1 which can be viewed as a constraint over thevariables with indices in S . Now there are two conditions:A Boolean function is a K -psudo-junta if(1) the expected number of variables in satisfied constraints isbounded by a constant K .(2) f (x) = f (y) if the variables in satisfied constraints and alsotheir values are the same for x and y .

Hatami’s theorem: For every C there is K (C ), such that if

I p(f ) < pC ,

then f is close to a K (C )-pseudo-junta.

Gil Kalai Discrete Isoperimetry

A conjectural extension of Hatami’s theorem

Conjecture: Suppose that µp(f ) = t and

I (f ) ≤ C log(1/t)t

then f is close to a O(log(1/t))-pseudo-junta.

Gil Kalai Discrete Isoperimetry

Stability versions of Harper’s theorems

Gil Kalai Discrete Isoperimetry

Symmetry: invariance under transitive group

Theorem: If a monotone Boolean function f with n variables isinvariant under a transitive group of permutation of the variables,then

I p(f ) ≥ Cµp(f )(1− µp(f )) log n.

Proof: Follows from KKL’s theorem since all individual influencesare the same.

Gil Kalai Discrete Isoperimetry

Total influence under symmetry of primitive groups

For a transitive group of permutations Γ ⊂ Sn, let I (Γ) be theminimum influence for a Γ-invariant function Boolean functionwith n variables.Theorem: [Bourgain and K. 1998] If Γ is primitive then one of thefollowing possibilities hold.

I

I (Γ) = θ(√

n),

I

(logn)(k+1)/k−o(1) ≤ I (Γ) ≤ C (log n)(k+1)/k ,

I I (Γ) behaves like (log n)µ(n), where µ(n) ≤ log log n isgrowing in an arbitrary way.

Gil Kalai Discrete Isoperimetry

Short Interlude: Kruskal-Katona for colorful (completelybalanced) and flag complexes

Frankl-Furedi-K.(88); Frohmader (2008)

Gil Kalai Discrete Isoperimetry

An important consequence of Frankl-Furedi-K. Theorem

I have Frankl number 1!

Gil Kalai Discrete Isoperimetry

An important consequence of Frankl-Furedi-K. Theorem

I have Frankl number 1!

Gil Kalai Discrete Isoperimetry

Part III: Kruskal-Katona for embedded complexes

Conjecture [high-dimensional crossing conjecture] (KarnabirSarkaria and K. late 80s) Let K be a d-dimensional simplicialcomplexes that can be embedded into 2d-dimensional space. Then

fd(K ) ≤ C (d)fd−1(K ).

Euler’s theorem asserts that C (1) = 3 it is conjectured thatC (d) = d + 2.

Gil Kalai Discrete Isoperimetry

The high-dimensional crossing conjecture - a known case

Theorem (K. early 90) Let K be a d-dimensional simplicialcomplexes that can be embedded as a subcomplex into theboundary complex of a 2d + 1-dimensional polytope. Then

fd(K ) ≤ (d + 2)fd−1(K ).

Gil Kalai Discrete Isoperimetry

Shifting

A combinatorial method initiated by Erdos Ko Rado to move froma general set system (or abstract simplicial complex) to a “shifted”one.

A shifted family of subsets of 1, 2, . . . , n is shifted if wheneveryou start with a set S in the family and replace an element j ∈ Swith an element i 6= S with i < j , then the set R you obtained alsobelongs to the family.

Example: Give i a real weight wi . Suppose thatw1 > w2 > · · · > wn. Let F be the family of all sets where thesum of weights is positive. Then F is shifted.

Shifting is an operation that replace a family of sets by a shiftedfamily of the same cardinality such that various combinatorialproperties are preserved.

Gil Kalai Discrete Isoperimetry

Symmetrization

The role of shifting in discrete isoperimetry is similar to the role ofsymmetrization in classical isoperimetry.

Gil Kalai Discrete Isoperimetry

Algebraic shifting

Algebraic shifting is an algebraic version of the Erdos-Ko-Radoshifting operation that I developed in my Ph. D. thesis and it wasstudied by Anders Bjorner and me in the late 80s and early 90s. Itallows to consider homological and algebraic properties of K .

Algebraic shifting maps a general simplicial complex K to a shiftedsimplicial complex ∆(K ), with the same f -vector.

Gil Kalai Discrete Isoperimetry

The high dimensional crossing conjecture - shifting version

Conjecture: Let K be a d dimensional simplicial complex whichcannot be embedded into R2d . Then ∆(K ) does not contain thed-skeleton of a 2d + 2-dimensional simplex.For d = 1 the conjecture asserts that ∆(K ) does not contain K5 acomplete graph on 5 vertices. This is known.

Gil Kalai Discrete Isoperimetry

Recent approach: (Wegner - Nevo)

Theorem (Uli Wagner): The weak version of the high-dimensionalcrossing conjecture holds for random simplicial complexes (in theLinial-Meshulam model).Moreover, the result holds if we exclude any subcomplex as a“minor” for a suitable definition of a term “minor”.

Gil Kalai Discrete Isoperimetry

Another related recent approach: Balanced complexes andbalanced shifting

This is an attack on the problem by restriction our attention tocompletely balanced complexes: d-complexes whose vertices canbe colored by (d+1) colors so that all simplices are rainbowsimplices. (Nevo, Novik, K., Babson, Wegner..)

Gil Kalai Discrete Isoperimetry

Gil Kalai Discrete Isoperimetry

Conclusion

Isoperimetric relations are important in several areas ofcombinatorics, extremal; probabilistic, geometric and algebraic, andare relevant to connections between combinatorics and other areas.

Thank You!

Gil Kalai Discrete Isoperimetry